Falling Factorial of Complex Number as Summation of Unsigned Stirling Numbers of First Kind

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Theorem

Let $z \in \C$ be a complex number whose real part is positive.

Then:

$z^{\underline r} = \displaystyle \sum_{k \mathop = 0}^m \left[{r \atop r - k}\right] \left({-1}\right)^k z^{r - k} + \mathcal O \left({z^{r - m - 1} }\right)$

where:

$\displaystyle \left[{r \atop r - k}\right]$ denotes the extension of the unsigned Stirling numbers of the first kind to the complex plane
$z^{\underline r}$ denotes $z$ to the $r$ falling
$\mathcal O \left({z^{r - m - 1} }\right)$ denotes big-$\mathcal O$ notation.


Proof


Historical Note

Donald E. Knuth acknowledges the work of Benjamin Franklin Logan in his The Art of Computer Programming: Volume 1: Fundamental Algorithms, 3rd ed. of $1997$.


Sources